Ab initio calculation of lattice dynamics and thermodynamic properties ofberylliumFen Luo, Ling-Cang Cai, Xiang-Rong Chen, Fu-Qian Jing, and Dario Alfè Citation: J. Appl. Phys. 111, 053503 (2012); doi: 10.1063/1.3688344 View online: http://dx.doi.org/10.1063/1.3688344 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i5 Published by the American Institute of Physics. Related ArticlesMetal induced crystallization mechanism of the metal catalyzed growth of silicon wire-like crystals Appl. Phys. Lett. 99, 143102 (2011) Note: Extraction of hydrogen bond thermodynamic properties of water from dielectric constant and relaxationtime data J. Chem. Phys. 135, 086101 (2011) Waste-recycling Monte Carlo with optimal estimates: Application to free energy calculations in alloys J. Chem. Phys. 135, 044127 (2011) Particle-based multiscale coarse graining with density-dependent potentials: Application to molecular crystals(hexahydro-1,3,5-trinitro-s-triazine) J. Chem. Phys. 135, 044112 (2011) Stacking in sediments of colloidal hard spheres J. Chem. Phys. 135, 034510 (2011) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Ab initio calculation of lattice dynamics and thermodynamic propertiesof beryllium

Fen Luo,1,2 Ling-Cang Cai,1,a) Xiang-Rong Chen,2,3,b) Fu-Qian Jing,1,2 and Dario Alfe4

1National Key Laboratory for Shock Wave and Detonation Physics Research, Institute of Fluid Physics,Chinese Academy of Engineering Physics, Mianyang 621900, China2College of Physical Science and Technology, Sichuan University, Chengdu 610064, China3International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, China4Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT,United Kingdom

(Received 17 October 2011; accepted 18 January 2012; published online 1 March 2012)

We investigate the phase transition, elastic constants, phonon dispersion curves, and thermal

properties of beryllium (Be) at high pressures and high temperatures using density functional

theory. By comparing the Gibbs free energy, in the quasiharmonic approximation (QHA), of

hexagonal-closed-packed (hcp) with those of the face-centered cubic (fcc) and body-centered-cubic

(bcc) we find that the hcp Be is stable up to 390 GPa, and then transforms to the bcc Be. The calcu-

lated phonon dispersion curves are in excellent agreement with experiments. Under compression,

the phonon dispersion curves of hcp Be do not show any anomaly or instability. At low pressure

the phonon dispersion of bcc Be display imaginary along C-N in the T1 branches. Within the quasi-

harmonic approximation, we predict the thermal equation of state and other properties including

the thermal expansion coefficient, Hugoniot curves, heat capacity, Gruneisen parameter, and Debye

temperature. VC 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3688344]

I. INTRODUCTION

Beryllium has a wide range of applications in aircrafts,

spacecrafts, communication satellites, nuclear power indus-

try and so on.1 In addition, Be can also be alloyed with other

elements to produce strong lightweight alloys for computer

parts and instruments, and other applications.2 For those im-

portant applications, scientific investigations on structural

stability and phase diagram of beryllium have attracted tre-

mendous experimental and theoretical interest.3–14

At ambient condition, Be has the hexagonal-closed-

packed (hcp) structure and melts at 1560 K.3 It transforms to

the body-centered-cubic (bcc) structure at 1523 K.4 Experi-

mentally, Ming et al.5 found the existence of a distorted hcp

phase at pressures between 8.6 and 14.5 GPa. Nakano et al.6

investigated the static compression of Be by x-ray diffraction

and found that the hcp phase was stable up to 171 GPa at

300 K. The hcp Be was found to be stable to the highest pres-

sure of 66 GPa by Velisavljevic et al.7 Evans et al.8 con-

firmed the stability of hcp Be to 200 GPa at 300 K with x-ray

and Raman studies in helium and argon pressure media. The-

oretically, Lam et al.9 predicted that the stable phase is either

bcc or face-centered-cubic (fcc) between 100 and 200 GPa

using the pseudopotential method. Meyer-ter-Vehn et al.10

showed that hcp Be transformed directly to bcc Be at 200

GPa based on the augmented spherical waves method. Later,

Palanivel et al.11 obtained that the lowest energy phase is

bcc Be above 180 GPa, using the full-potential linear muffin-

tin orbital (FP-LMTO) method. Sin’ko et al.12 used the FP-

LMTO method and discovered that hcp-bcc phase transition

pressure is 270 GPa at 300 K. Moreover, the phase bounda-

ries of the hcp-bcc phase transition were calculated by Bene-

dict et al.13 and Robert et al.,14 and they all predicted the hcp

phase as the most stable structure up to 400 GPa at 0 K.

Although Be has a simple atomic structure, it remains chal-

lenging to accurately determine its phase transitions.

It is important to have a systematic study of the thermal

equation of state (EOS) to understand the phase diagram and

dynamical response of materials under extreme conditions.

The elastic properties of materials at high pressure are im-

portant because they relate to various fundamental physicals

properties such as mechanical strength, and the EOS. The

elastic constants have been measured at ambient condition15

or high temperature.16,17 Based on density-functional theory,

the elastic constants of Be have been investigated by several

groups.2,12,18 However, the investigations of thermal EOS

and elastic properties of Be under extreme conditions are still

scarce. In the present work, we focus on the thermodynamic

and elastic properties of Be under high pressure.

In a neutron scattering experiment,19 the low-temperature

phonon dispersion relation of the hcp phase of Be did not

show any anomaly or instability at ambient pressure. Theoret-

ically, combining frozen phonon and density-functional per-

turbation theory methods, Robert et al.20,21 obtained the

phonon dispersion relations of hcp Be and bcc Be at different

densities. They have been successful at reproducing the exper-

imental investigation and found that the hcp phase is stable

from 1.4 to 6 g/cc. At ambient conditions, the phonon disper-

sion relation of bcc Be exhibited a large softening near the Npoint of the T1 branch along the C-N direction, which showed

that the bcc phase is unstable. The frequency near 2/3 [111]

transverse mode is also anomalous low when the density

decreases under 1.65 g/cc. One of the main purposes of this

a)Electronic mail: cai_lingcang@yahoo.com.cn.b)Electronic mail: xrchen@126.com.

0021-8979/2012/111(5)/053503/10/$30.00 VC 2012 American Institute of Physics111, 053503-1

JOURNAL OF APPLIED PHYSICS 111, 053503 (2012)

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work is to investigate the lattice dynamics and thermodynam-

ics of Be under high pressure and temperature.

Here, we apply density functional perturbation theory

(DFPT)22,23 to investigate a wide range of properties, includ-

ing the lattice dynamical properties, the thermal EOS and

thermodynamic properties of Be under high pressure. First,

we discuss the structural and elastic properties of Be under

high pressure. Then, we study the phase diagram and the

thermodynamic properties within the quasiharmonic approx-

imation (QHA).

II. COMPUTATIONAL METHOD

According to QHA, the free energy is given by

FðV; TÞ ¼ EstaticðVÞ þ FphonðV; TÞ þ FelecðV; TÞ; (1)

where Estatic (V) is the energy of a static lattice at zero tem-

perature T and volume V, Felec(V,T) is the thermal free

energy arising from electronic excitations, and Fphon(V,T) is

the phonon contribution. Both Estatic (V) and Felec(V,T) can

be evaluated via static first-principles calculations directly.

The phonon vibrational contribution Fphon(V,T) can be

expressed as

FphonðV; TÞ ¼1

2

Xq;j

�hxj q;Vð Þ

þ kBTX

q;j

ln 1� exp ��hxj q;VÞ=kBTð Þ� �� �

;

(2)

where kB is the Boltzmann constant, � is the Plank constant

divided by 2 p, and xj(q,V) is the phonon frequency of the

jth mode of wave vector q in the first Brillouin zone (BZ).

The phonon dispersion calculations of Be have been per-

formed within DFPT as implemented in the QUANTUM

ESPRESSO package24 using local-density approximation

(LDA)25 with the parametrization of Perdew and Zunger.26

A nonlinear core correction to the exchange-correlation

energy function was introduced to generate a norm-

conserving pseudopotential27 for Be with the valence elec-

trons configuration 2s2. The pseudopotential was generated

with a nonrelativistic calculation using the LDA.

To ensure the convergence of phonon frequencies and

free energies, we made careful tests on k and q grids, the ki-

netic energy cutoff, and smearing parameters. For hcp Be, dy-

namical matrices were computed at 28 wave (q) vectors using

an 6� 6� 6 q grid in the irreducible wedge of the Brillouin

zone. The plane wave cutoff for the wave functions was 60

Ry. The Monkhorst-Pack (MP)28 meshes were 20� 20� 14

for hcp Be and 20� 20� 20 for bcc Be in both total energy

and phonon dispersion calculations, respectively. We com-

puted the dynamical matrices at 29 wave (q) vectors using the

8� 8� 8 q grid for bcc Be in the irreducible wedge of the BZ,

and the full phonon dispersion was obtained through Fourier

interpolation. We applied a Fermi-Dirac smearing width of

0.02 Ry. The geometric mean phonon frequency is defined by

ln �x ¼ 1

Nqj

Xqj

ln �xqj; (3)

where xqj is the phonon frequency of the branch j at the

wave vector q and Nqj is the number of branches times the

total number of q points in the sum. With this choice of pa-

rameters, the geometric mean phonon frequency x was con-

verged to 1 cm�1.

The elastic constants are defined by means of a Taylor

expansion of the total energy, E (V, d), for the system with

respect to a small strain d of the lattice primitive cell volume

V. The energy of a strained system is expressed as follow

EðV; dÞ ¼ EðV0; 0Þ þ V0

Xi

sinidi þ1

2

Xij

Cijdinjdj

" #; (4)

where E(V0, 0) is the energy of the unstrained system with

equilibrium volume V0, si is an element in the stress tensor,

and ni is a factor to take care of the Voigt index.29

The elastic constants of Mo30 and Ti31 were successfully

obtained using the theoretical method proposed by Sin’ko

and Smirnov.32 Here we give a description of this method.

The lattice vectors a0 of the strained primitive cell are deter-

mined from the lattice vectors a of the equilibrium primitive

cell by the relation a0 ¼ aðI þ e_

iÞ, where I is the unit matrix

and e_

i is a strain tensors. To calculate five independent con-

stants of hexagonal structures, we considered five independ-

ent volume-nonconserving strains

e_

1 ¼c 0 0

0 0 0

0 0 c

0B@

1CA; e

_

2 ¼c 0 0

0 �c 0

0 0 0

0B@

1CA;

e_

3 ¼c 0 0

0 c 0

0 0 0

0B@

1CA; e_4 ¼

0 0 c

0 0 0

c 0 0

0B@

1CA; e

_

5 ¼0 0 0

0 0 0

0 0 c

0B@

1CA:(5)

The specific energy of the crystal deformed in accord with

the e_

1 was calculated as a function of the strain magnitude c.

The values of strain c with the range of 65% were used to

make the strain energy fit to a second-order polynomial func-

tion. Then the strain energy as a function of the strain can be

written as

C11 þ 2C13 þ C33 � 2P ¼ q1

d2Eðq1; e_

1Þdc2

�����c¼0

; (6)

where P was the pressure applied. Similar results with the

other matrix were used in the equation

2ðC11 � C12 � PÞ ¼ q1

d2Eðq1; e_

2Þdc2

�����c¼0

; (7)

2ðC11 þ C12 � PÞ ¼ q1

d2Eðq1; e_

3Þdc2

�����c¼0

; (8)

4C44 � 2P ¼ q1

d2Eðq1; e_

4Þdc2

�����c¼0

; (9)

053503-2 Luo et al. J. Appl. Phys. 111, 053503 (2012)

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C33 � P ¼ q1

d2Eðq1; e_

5Þdc2

�����c¼0

: (10)

III. RESULTS AND DISCUSSION

A. Structural and elastic properties

We calculated the static energy-volume (E-V) curves for

hcp, bcc, and fcc structures. Figure 1(a) shows the static

energy of the hcp, bcc, and fcc as a function of volume. The

static energy differences are plotted in the inset of Fig. 1(a).

For the applied wide range of volume, the static energy dif-

ference between fcc and hcp is positive. By fitting the E-Vdata to the fourth-order finite strain EOS,33 we calculated the

enthalpy versus pressure for the three structures at zero tem-

perature [Fig. 1(b)]. The transformation from hcp to bcc

occurs at 390 GPa, in good agreement with experiments,6–8

and theoretical calculations.13,14 Therefore, in the present

work, all calculations were performed for the hcp phase. The

equilibrium volume V0, lattice parameters a and c, the bulk

modulus B0, and its pressure derivative B0, are list in Table I.

Our results are also in excellent agreement with the available

experimental data.34–37 We note that the bulk modulus of

hcp Be at 0 K is 122 GPa, and decreases to 118 GPa when

thermal effects are included.

The elastic constants of hcp Be at each volume were cal-

culated using the method discussed above. The pressure was

obtained from the static equation of state. Table II summaries

the calculated elastic constants at equilibrium structure param-

eters, together with the experimental data15 and previous cal-

culations.12,20 Our results agree well with both sets of data.

From Table II, one can see that all the elastic constants are

practically linear increasing as pressure increases. Conse-

quently, we fit the calculated elastic constants as the form

Cij(P)¼ aijþ bijP, and tabulated the fitting parameters in

Table III. Recently, Cazorla et al.38 have found similar trends

in hcp He using atomistic Diffusion Monte Carlo calculations

that fully account for ionic quantum effects. The errors of all

elastic constants are �4% in our results, which due to the

uncertainties of the energies and corresponding polynomial

fits. For a stable hexagonal structure, its five independent elas-

tic constants Cij should satisfy the mechanical stability criteria,

i.e., ~C44 > 0, ~C11 > ~C12

�� ��, and ~C33~C11 þ ~C12

� �> 2 ~C2

13, where~Caa ¼ Caa � P a ¼ 1; 3; 4ð Þ, ~C12 ¼ C12 þ P, ~C13 ¼ C13 þ P.

Clearly, these calculated elastic constants Cij satisfy the

mechanical stability criteria, suggesting that hcp Be is mechan-

ically stable under the applied pressure.

Based on the Voigt-Reuss-Hill approximation,39 we have

calculated the corresponding bulk, shear and Young’s modu-

lus from the single crystal elastic constants. Then the isotropic

averaged aggregate velocities can be obtained as follows

VP ¼ Bþ 4=3Gð Þ=q½ �1=2; (11)

VB ¼ ðB=qÞ1=2; (12)

FIG. 1. (a) Static energy per atom of the bcc, fcc, and hcp phases as function of volume. (b) Enthalpy per atom of the bcc and fcc phases relative to the bcc

enthalpy as function of pressure.

TABLE I. The equilibrium volume V0 (A3/atom), lattice parameters a and

c, axial ratio c/a, zero pressure bulk modulus B0 (GPa), and pressure deriva-

tive B0.

V0 a c c/a B0 B0

hcp present 7.723 2.248 3.529 1.57 122 3.29

Present* 7.910 2.266 3.556 1.569 118 3.21

experimental 8.10534 2.28534 3.58534 1.56934 11836 3.5236

experimental 8.11035 2.28635 3.58435 1.56835 11937 3.4837

bcc Present 7.665 2.484 120 3.34

fcc Present 7.710 3.136 118 3.33

The asterisk * indicates the values obtained from 300 K EOS.

053503-3 Luo et al. J. Appl. Phys. 111, 053503 (2012)

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VS ¼ ðG=qÞ1=2; (13)

where VP, VB, and VS are the compressional, bulk, and shear

sound velocities, respectively. In addition, as listed in Table

II, the bulk modulus B deduced from elastic constants turns

out to be very close to that obtained by EOS fitting. This

indicates that our calculations are consistent and reliable.

The bulk modulus, shear modulus, Young’s modulus, and all

the sound velocities increase monotonously with pressure

increasing, as presented in Figs. 2(a) and 2(b).

It is known that the acoustic velocities are related to the

elastic constants by the Christoffel equation40

Cijklnjnk �Mdil

� �ui ¼ 0; (14)

where Cijkl is the fourth-rank tensor description of the elastic

constants, n is the propagation direction, q is the density, and

u is the polarization vector. The acoustic anisotropy can be

described as41

Di ¼Mi nx½ �

Mi 100½ � ; (15)

where nx is the extremal propagation direction, and i denotes

the three types of elastic waves (one longitudinal and two

polarizations of the shear wave). By solving the Christoffel

equation for hcp Be, one can obtain the anisotropy of

the compressional wave (P), Dp¼C33/C11, and the anisotro-

pies of the wave polarized perpendicular to the basal plane

(S1) and the polarized one in the basal plane (S2), DS1

¼ (C11þC33 � 2C13)/4C44, DS2¼ 2C44/(C11 � C12).

The calculated pressure dependences of the anisotropies

Dp, DS1 and DS2 for the three types of elastic waves are illus-

trated in Table II. It is noted that with DS2 and Dp decrease as

pressure increases, while DS1 increases with the pressure

increasing. We also noted that as the pressure increases the

values of DS1 and DS2 goes away from 1.0, which indicates

that the anisotropies of the wave polarized perpendicular to

the basal plane (S1) and the wave polarized in the basal

plane (S2) become strong.

B. Phonon dispersions and phase transition

Within the DFPT, we obtained phonon dispersion curves

of hcp and bcc Be along high-symmetry directions. In Fig.

3(a), we compare our results with the experimental disper-

sion curves19 at 80 K. The agreement between the two sets

of data is impressive, and typical of DFPT. The hcp phase of

Be is dynamically stable at zero pressure as the phonon

TABLE II. The calculated elastic constants and anisotropies Dp, DS1, and DS2 for the three types of elastic waves of hcp Be, compared with the experimental

data and the other theoretical results. Cij, B, G, and E are in GPa.

P C11 C12 C13 C33 C44 B G E Dp DS1 DS2

0 310.9 19.5 19.1 359.5 162.1 121.6 155.3 326.7 1.156 0.975 1.113

25.11 487.03 34.95 26.13 553.56 233.42 188.9 234.3 497.3 1.137 1.059 1.033

50.02 643.8 50.37 32.45 724.04 296 248.9 303.7 647.8 1.125 1.100 0.998

75.04 792.43 65.48 39.31 890.23 353.07 306.8 368.6 789.6 1.123 1.136 0.971

100.03 935.27 80.78 46.54 1049.29 407.74 362.8 430.6 925.7 1.122 1.160 0.954

125.01 1072.31 97.62 54.09 1192.8 460.04 416.4 489 1054.3 1.112 1.172 0.944

150.15 1210.32 111.35 61.49 1353.04 511.17 471.1 548.7 1185.7 1.118 1.194 0.930

174.88 1342.38 127.04 68.99 1498.32 560.18 523.4 604.8 1309.9 1.116 1.206 0.922

200.41 1477.5 142.98 76.33 1643.9 609.61 576.4 661.8 1435.9 1.113 1.217 0.914

224.98 1606.3 158.8 84.75 1782.61 656.06 627.7 715.5 1555.6 1.110 1.227 0.906

249.98 1734.35 174.72 92.89 1935.06 702.45 680.2 769.9 1677 1.116 1.240 0.901

274.88 1861.42 190.7 101.58 2055.69 748.12 729.4 821.8 1792.3 1.104 1.241 0.896

299.9 1988.74 207.08 109.57 2215.09 791.86 782.5 875.1 1912.4 1.114 1.258 0.889

Exp. (Ref. 15) 293.6 26.8 14 356.7 162.2 116.8 150.1 315.2 1.215 0.959 1.216

Ref. 12 300.8 14.1 7.1 359.5 160.2 112.8 154.5 318.2 1.195 1.008 1.118

Ref. 20 305.9 18.8 10.4 329 159.3 114.5 152.4 316.7 1.076 0.964 1.110

TABLE III. Value of the parameters obtained in the linear fits to our Cij(P)

results. aij is given in units of GPa, and bij is given in GPa/K.

C11 C12 C13 c33 C44

aij 327.89 19.43 18.58 378.72 173.15

bij 5.78 0.61 0.29 6.36 2.18

FIG. 2. (a) Bulk modulus B, shear modulus G, and Young’s modulus E vs

pressure at zero temperature. (b) Aggregate sound velocities (VP, VB, and

VS) vs pressure at zero temperature.

053503-4 Luo et al. J. Appl. Phys. 111, 053503 (2012)

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frequencies do not show any anomaly. The calculated pho-

non dispersions for bcc Be are displayed in Fig. 3(b). Note

that the frequencies around N point along C� N in the trans-

verse acoustical branch soften to imaginary frequencies,

indicating dynamic instability.

To understand the dynamic stability of hcp and bcc Be at

high pressure, it is useful to investigate lattice vibrations at dif-

ferent volumes. Figure 4(a) shows that the frequencies in the

dispersion curves of hcp Be increase with decreasing volume.

We do not find soft modes in the applied range of volumes,

and the phonon frequencies reflect the dynamic stability. The

calculated phonon dispersions at different volumes for bcc Be

are displayed in Fig. 4(b), also for bcc, our results show that

the phonon frequencies increase as volume decreases. The cal-

culations predict the stability of the bcc phase of Be, and the

frequencies of the phonons along the C� N symmetry line and

around the N symmetry point are real.

In order to obtain accurately Helmholtz free energy F as

functions of volume V at a certain temperature, the F at 17

atomic volumes (between V¼ 3 and 8 A3/atom) for hcp and

bcc phases have been calculated. We determined the phase

transition pressure by comparing the Gibbs free energies of

hcp and bcc Be. The Gibbs free energy G is calculated from

the Helmholtz free energy as G(P,T)¼F(V,T)þPV. Within

QHA, the transition pressure is calculated at various temper-

atures up to 3500 K. The phase diagram of Be at high pres-

sure and temperature is shown in Fig. 5, compared with

theoretical calculation.21 It can be seen that the phase transi-

tion from hcp to bcc at 386 GPa at 300 K, which indicates

that at ambient temperature the hcp phase is more stable than

the bcc phase. Most experiments6–8 have not detected any

kind of phase transition up to 300 GPa. Robert et al.14,21

found that the hcp Be is the most stable phase up to 400

GPa. In Fig. 5, the dotted line is from the calculations of

Robert et al.,21 who obtained the phase diagram with the

QHA. Our transition pressures are consistent with those cal-

culated by Robert et al.21 at low temperatures, and show a

small deviation at high temperatures, which mainly due to

the choice of the LDA functional.

C. Thermodynamic properties

The Helmholtz free energy and phonon contribution

Fphon as functions of volume V and temperature T can be

derived from Eq. (1) and (2), respectively. The thermal free

energy arising from electronic excitations Felec and the pho-

non contribution Fphon as functions of volume at tempera-

tures from 300 to 3500 K are shown in Figs. 6(a) and 6(b).

By fitting the Helmholtz free energy to the fourth-order finite

strain EOS at each temperature, we get the theoretical iso-

thermal compressional curves, as shown in Fig. 7. We esti-

mate the errors of the Helmholtz free energy are less than

0.02%, which derived from the uncertainties of the energies

and corresponding EOS fits. Including the zero point energy

(ZPE) to the free energy at 0 K raises the equilibrium volume

by 1.9% and reduces the bulk modulus by 2 GPa relative to

the values without ZPE. It is noted that the 0 K isotherm

(including ZPE) is almost the same as the 300 K one, and

this is due to the small free energy contribution from the lat-

tice vibrations at 300 K. With the pressure increasing, our

isotherms agree with the experimental data.8

The thermal pressure can be obtained from the pressure

difference with the 0 K isotherm. The thermal pressure as

functions of volume and temperature are shown in Figs. 8(a)

and 8(b). From Fig. 8(a), one notes that the thermal pressures

show little volume dependence at low temperature. At ele-

vated temperature, the thermal pressure increases signifi-

cantly, and decreases monotonously with increasing volume.

FIG. 3. Phonon dispersion curves of (a) hcp Be, (b) bcc

Be at 0 GPa and 0 K. The solid circles in (a) are neutron

diffraction data (Ref. 19) measured at 80 K.

053503-5 Luo et al. J. Appl. Phys. 111, 053503 (2012)

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The thermal pressures as a function of temperature are

shown in Fig. 8(b). At a given volume, the thermal pressure

increases linearly with temperature. The slopes of the ther-

mal pressure show strong volume dependence. Many previ-

ous calculations for metals (such as Mo30 and Ta42 and so

on) also found that the slopes of the thermal pressure were

strongly volume dependent.

To check the accuracy of our thermal EOS at high pres-

sures, we calculated the Hugoniot P-V and P-T curves by

solving the Rankine-Hugoniot equation:

UH � U0 ¼1

2PH þ P0ð Þ V0 � VHð Þ; (16)

where UH, PH, and VH are the molar internal energy, pressure,

and volume along the Hugoniot states, respectively, and U0

and V0 are the molar internal energy and volume at zero pres-

sure P0 and room temperature. For a given volume, we

adjusted temperature and obtained pressure and energy until

the Rankine-Hugoniot relation was satisfied. We obtained the

pressure–volume and temperature–pressure relations along

the Hugoniot curve according to Eq. (16). Figure 9 shows the

obtained Hugoniot curve of P–V and P–T. Our obtained

Hugoniot curves are in excellent agreement with experi-

ments43 and other calculations.14,44,45 Our P-T curve agrees

well with the theoretical predictions of Robert et al.14 and

Holian et al.,44 and deviated from Song et al.45

FIG. 4. The phonon dispersion curves of the hcp (a) and bcc (b) Be shown along high-symmetry directions at different volumes.

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The thermal expansion coefficient aV is defined as:

aV ¼1

V

@V

@T

P

: (17)

The thermal expansion coefficient as functions of pressure

and temperature are plotted in Figs. 10(a) and 10(b). In Fig.

10(a), it is showed that our zero pressure results are in excel-

lent agreement with the experimental data46 at high tempera-

ture. The discrepancy between the calculated thermal

expansion coefficients and the experimental results at high

temperature might be attributed to the use of the LDA func-

tion. At ambient conditions, our calculated thermal expan-

sion coefficient is 3.43� 10�5 K�1, consistent with the

experimental value 3.4� 10�5 K�1.47 As the pressure rises,

the thermal expansion coefficient increases with temperature,

and then converges to a nearly constant value at high temper-

ature. The thermal expansion coefficient decreases strongly

with increasing pressure in Fig. 10(b).

The specific heat at constant volume is defined by

CV ¼@U

@T

V

; (18)

where U is the internal energy of the system. The specific

heat at constant pressure CP is different from CV due to the

FIG. 5. Phase diagram of Be at high pressure and temperature. The dotted

line is the hcp-bcc boundary calculated by Robert et al. (Ref. 21), the dashed

line is the theoretical melting curve by Robert et al. (Ref. 21).

FIG. 6. Free energy from the phonons

Fphon (a) and electronic excitations Felec

(b) vs volume of hcp Be at temperatures

from 300 to 3500 K.

FIG. 7. Isothermal compression curves at different temperatures, compared

with experimental data (Refs. 6, 8).

053503-7 Luo et al. J. Appl. Phys. 111, 053503 (2012)

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thermal expansion caused by anharmonic effects. The rela-

tionship between CP and CV is given by

CP � CV ¼ a2V Tð ÞB0VT; (19)

where aV is the volume thermal expansion coefficient, and B0

the bulk modulus. In a harmonic crystal CV increases at low

temperature and approaches the constant 3R at high tempera-

ture. In Fig. 11, we plot CP, which decreases with pressure,

and increases with temperature. The available experimental

results48 at zero pressure are also plotted as squares for com-

parison. It is evident that our QHA results are in overall good

agreement with the experimental data at zero pressure, but

diverge at high temperature. It is found that Cp is obvious dif-

ferent between low and high pressures with increasing tem-

perature beyond 1600 K (about the melting temperature),

which implies that validity of QHA cannot be extended to

high temperature at low pressure and we should consider

higher-order anharmonicities. The melting temperature

increases with pressure, and since the QHA is only valid up to

some fraction of the melting temperature, then the validity of

the QHA extends to higher temperatures at higher pressures.

The thermodynamic Gruneisen parameter is a very im-

portant parameter through which the thermal pressure is

related to the increase of thermal energy in the Mie–Gruneisen

equation of state and it is defined by

c ¼ V@P

@U

V

¼ aKTV

CV; (20)

where U is the internal energy of the system. Our calculated

zero-pressure 300 K value of c is 1.271, very close the experi-

ment value 1.203.15 The value of c versus temperature and

pressure are shown in Figs. 12(a) and 12(b), respectively. From

Fig. 12(a), it is obvious that c shows weak dependence on tem-

perature along the isobar. Upon compression, c decreases sig-

nificantly, but the temperature effects become less pronounced

[Fig. 12(b)]. In addition, the values of c at high pressures

remain nearly constant.

Within the Debye approximation, the Helmholtz free

energy at low temperature is

F ¼ Estatic þ RT9

8

HD

T

þ 3 ln 1� e�HD=T

� �� D

HD

T

� �� �;

(21)

D HD=Tð Þ is the Debye function written as

D HD=Tð Þ ¼ 3T

HD

3ðHD=T

0

z3dz

ez � 1: (22)

We obtained HD at temperature T by solving Eq. (21). Our

results for the Debye temperature are displayed in Fig. 13. HD

shows a significant increase as pressure increases (decrease of

volume). At fixed volume, HD drops with increasing T up to

�100 K, and then decreases moderately in the temperature

FIG. 8. Calculated thermal pressures of hcp Be (a) as a function of volume

and (b) temperature.

FIG. 9. Volume–pressure (a) and temperature–pressure (b) relations on

Hugoniot curves obtained from the QHA, in comparison with experimental

data (Ref. 43) and other calculations (Refs. 14, 44, 45).

053503-8 Luo et al. J. Appl. Phys. 111, 053503 (2012)

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range from around 100 to 300 K. With further increase of T

(above 300 K), HD shows weak temperature dependence,

especially at high pressure. At 300 K and 0 GPa, the obtained

value of HD is 930 K, which is consistent with the experimen-

tal values of 949 6 51 K.49–51

IV. CONCLUSIONS

We employed the density functional perturbation theory

to investigate the phase transition, elastic properties, lattice

dynamical properties, and thermodynamic properties of the

hcp structure of Be. By comparing the static energy-volume

and enthalpy-pressure curves for the hcp, bcc, and fcc Be,

we found that the stable phase of Be is hcp at zero pressure

and temperature and then transforms to bcc Be at 390 GPa.

The fcc phase is never be the stable phase in the whole range

of pressures. At ambient conditions, the calculated elastic

constants agree well with the experimental data. From the

elastic constants at high pressure, we found that hcp Be is

stable. Bulk, shear, and Young’s modulus, and all the sound

velocities as a function of pressure are obtained.

FIG. 10. Thermal expansion coefficient aV as a function of temperature (a)

and pressure (b). The solid squares and solid triangle are taken form Gordon

et al. (Ref. 46) and Grimvall et al. (Ref. 47), respectively.

FIG. 11. Heat capacity CP as a function of temperature at different pressure,

together with the experimental data (Ref. 48).

FIG. 12. Variation of the Gruneisen parameter c with temperature (a) and

pressure (b).

FIG. 13. Debye temperature HD as a function of temperature at different

volumes.

053503-9 Luo et al. J. Appl. Phys. 111, 053503 (2012)

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Our calculated phonon dispersion curve of hcp Be at zero

GPa agree extremely well with experiment. Under compres-

sion, the dispersion curves of hcp Be do not show any anom-

aly or instability. The frequencies of bcc Be along C-N in the

T1 branches soften to imaginary frequencies at zero pressure,

indicating a structural instability, while with pressure increas-

ing, the imaginary frequencies of bcc Be along C-N in the T1

branches shift to positive frequencies. With the QHA, the

hcp-bcc boundary under high temperature and pressure was

obtained. Our results agree well with those reported by Robert

et al. We predicted the thermal EOS properties including ther-

mal EOS, thermal pressure, volume thermal expansion, and

Hugoniot properties. The zero-pressure–temperature depend-

encies of volume thermal expansion coefficient, Gruneisen

parameter, and Debye temperature are found to be in a good

agreement with the experimental results. The QHA is only

valid up to a fraction of the melting temperature, and therefore

results can only expected to agree with experiments at suffi-

ciently low temperatures. Since the melting temperature

increases with pressure, the validity of the results extends to

higher temperatures with increasing pressure.

ACKNOWLEDGMENTS

The authors would like to thank Dr. Zhao-Yi Zeng for

the helpful discussions. We also thank the support by the

National Natural Science Foundation of China under Grant

Nos. 10776029/A06 and 11174214, the Specialized Research

Fund for the Doctoral Program of Higher Education under

Grant No. 20090181110080 and the Science Foundation of

China Academy of Engineering Physics under Grant No.

2010A0101001.

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